Let f(x) be a function such that f'(x)=g(x) and f''(x)=-f(x). Let h(x)=\{f(x)\}^{2}+\{g(x)\}^{2}. Then consider the following statements : 1. h'(3)=0 2. h(1)=h(2) Which of the statements given above is/are correct?
- A. 1 <strong>ONLY</strong>
- B. 2 <strong>ONLY</strong>
- C. <strong>BOTH</strong> 1 and 2 ✓
- D. <strong>NEITHER</strong> 1 nor 2
Correct Answer: C. <strong>BOTH</strong> 1 and 2
Explanation
Differentiate h(x) with respect to x: h'(x) = 2f(x)f'(x) + 2g(x)g'(x). Substitute f'(x) = g(x) and g'(x) = f''(x) = -f(x). We get h'(x) = 2f(x)g(x) + 2g(x)(-f(x)) = 0. Since h'(x) = 0 for all x, h(x) is a constant function. Therefore, h'(3) = 0 is true, and h(1) = h(2) is also true.
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